The chamber complex for the Littlewood-Richardson coefficients of GL4

THE CHAMBER COMPLEX FOR THE LITTLEWOOD-RICHARDSON COEFFICIENTS OF GL4. EMMANUEL BRIAND, MERCEDES ROSAS, AND STEFAN TRANDAFIR The Li lewood-Richa dson coe icien s cν λ,µ a e a amily o cons an s o cen al impo ance in ep esen a ion heo y. They a e indexed by iples o in ege pa - i ions. I is known ha when he size o he pa i ions is limi ed o some ixed in ege N(coe icien s associa ed o GLN), hese coe icien s a e gi en by piecewise polynomial o mulas in λ1,...λN, µ1, . . . , µN, ν1, . . . , νN−1. Mo e p ecisely, he do- mains o polynomiali y a e he big cells (”chambe s”) o a subdi ision o a cone in R3N−1( he ”chambe complex”). In 2005, ´ E. Rassa has calcula ed explici ly he chambe complex o N= 3, inding 18 chambe s subdi iding a cone in R8, and he co esponding polynomial o mulas (in his case, linea ). Using Rassa ’s desc ip ion, E. B iand and M. Rosas ecen ly lis ed exhaus i ely all symme ies o he chambe complex o N= 3, inding a g oup o 288 symme ies (much mo e ha he 24 symme ies known o exis o gene al N). I is na u al o look a wha happens in he nex case, N= 4. Wi h his aim, we ha e compu ed (using Sagema h and Singula ) he chambe complex o N= 4. This objec is la ge, since i consis s in he subdi ision o a cone in R11 in 67769 chambe s. This compu- a ion is based on known combina o ial desc ip ions o he Li lewood-Richa dson coe icien s. Fo ins ance, he Li lewood-Richa dson coe icien c(26,21,14,11) (16,13,5,2),(17,12,6,1) coun s he iangles 36 34 53 29 x365 16 x1x271 0 26 47 61 72 illed wi h in ege s subjec o he inequali ies a+b≥c+din all hombi: d b c a b c b d a c a d o he iangle. (The bo om bo de en ies 0,26,47,61,72 a e he cumula ed sums o (26,21,14,11), ha a e: 0,26,26+21,26+21+14,26+21+14+11 and he op bo de en ies 0,16,29,34,36,5365,71,72 a e he cumula ed sums o (16,13,5,2) and (17,12,6,1), ha a e 0,16,16+13,16+13+5,...,16+13+5+2+17+12+6+1). E. B iand, Depa amen o Ma em´ a ica Aplicada I, Uni e sidad de Se illa M. Rosas, Depa amen o de ´ Algeb a, Uni e sidad de Se illa S. T anda i , Simon F ase Uni e si y. E. B iand and M.Rosas a e pa ially suppo ed by MTM2016-75024-P and FEDER, and Jun a de Andalucia unde g an FQM333.